A complex sign Doppler estimator (CSDE) is proposed for real-time hardware-based synthetic aperture radar (SAR) Doppler centroid estimation. The CSDE reduces the complexity in Doppler centroid estimation by exploiting the statistical properties of SAR data and applying the complex arcsine law to the correlation coefficient calculation. In fact, SAR data approximates the static complex Gaussian process, so fast estimating techniques for the angle of the normalized correlation can be applied to the estimating process. The proposed new estimator requires only the symbols of the imaginary and real component of the raw data while computing the phase. As a result, very simple estimating arithmetic is needed for the whole computation procedure. Analytical and simulation results show that with the CSDE, not only much less computational cost and hardware implementation area can be achieved, but also comparable performance with the known sign Doppler estimator (SDE) can be maintained.